The why's of the wing design

All was done to keep the lines smooth and the weight low.

Burt Rutan already had the wings on paper. Two pair of wings with nearly the same span. The front wings are pointing downwards (anhedral is the right technical word). The rear wings are pointing upwards (dihedral is the right word here).

Why two nearly identical wings?

Well, I guess that Mr. Rutan wanted to avoid a heavy wing. So he choose two short wings. A short wing does not have to be reinforced allover to prevent the wing from breaking. A short wing can be made lighter. Just compare gliders with biplanes to understand. The better performing loooong wings of a glider look fragile. They are! They need special reinforcements to keep them in one piece during flight. Biplanes are not so efficient with their short wings, but they can be very light. OK, the stings between the wings give extra support. But biplanes without designed without those strings are light too. There is less chance for these short wings to break in the air.

Why those up and down pointing wings?

Bob Walters, the designer of the Dragonfly, said to Jack Lambie in Jacks book "Composite Construction for homebuilt aircraft" (ISBN: 0-938716-26-3):

"No one has ever tested this idea in a wind tunnel but keeping the wing tips separated may reduce the interference drag."

With other words ... the front wings position would interfere less with the performance of the rear wing if the wing tips were not in the same horizontal line or even close.

I recently read that there was also another reason which i never expected. In Homebuilairplanes forum i read the follwing: Cluttonfred wrote on dec 13 2023: " I believe the front wing anhedral/rear wing dihedral on the Quickie was motivated by the wingtip-mounted main gear and a desire to look like an X-Wing from Star Wars (seriously)." Some other member asked: "Just how fast did they design and build the Quickie? First flight was only about 6 months after the movie was released." Hot Wings answered with a old article that mentions that the Quicky prototype construction began on aug 13 1977, took 400 hours over a period of 3 months. 
So ... this StarWars reason might have been right. Haha ... i just loooove aviation history details like this one.

Why those wheels in the wing tips?

Here you can see the genius inside Burt Rutan. If he wanted to place a classic landing gear (two wheels in front and one near the tail) to save weight, he still had to place two long legs to prevent the wing tips from scraping the ground during ground maneuvers. Mr. Rutan must have thought: "Hey, I already have two long legs. The front wings!" OK, everybody can imagine that placing wheels there would test the wings harder during landings. Those front wings needed to be reinforced! Burt Rutan probably found out that those reinforcements were less in weight than the otherwise needed long legs.

So Burt Rutans design lost some weight AND preserved its superb smooth lines.